#9865896

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(3,8)(6,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,29,32)(27,33,35,30)(28,31,37,36)(34,39,40,38);(1,2,5,4)(3,7,8,6)(12,19)(16,22)(18,23)(20,24)(25,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(10,15)(11,17)(16,22)(18,23)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,2,5,4)(3,7,8,6)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9773092);